# Gauge theory of gravity

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Moderator's note: Spin off from previous thread due to advanced nature of topic.

If we consider GR as an "effective field theory", i.e., as a low energy approximation to some underlying quantum theory, then yes, we can consider gravity as an "interaction" on that view. That is the view that Weinberg, for example, is advocating in the article you linked to. But that view is off topic in this forum; discussion of it belongs either in the quantum physics forum, or more likely in the Beyond the Standard Model forum since that's where discussion of quantum gravity in general belongs. See my post #3 for further discussion of this point.
There is classical field theory too, and GR is a relativistic classical field theory of the gravitational interaction. It's ironic that you fight for a geometrical-interpretation-only point of view and at the same time forbid the discussion of a very strong argument for this point of view: Starting from the gauge-theoretical approach (making Poincare transformations local) you end up inevitably with the geometrical point of view. If you restrict yourself to the classical description of spin-saturated matter and electromagnetism you even end up with GR proper, i.e., a pseudo-Riemannian (Lorentzian) manifold. Including spin you are lead to Einstein Cartan theory, i.e., a spacetime manifold with torsion, which is however still a theory where the gravitational interaction is geometrized as in GR, it's only a bit more comprehensive.

All this has nothing to do with quantum field theory. The question, whether a future consistent quantum theory of gravitation will be a QFT or based on some more general concept, is of course wide open, and discussions/speculations about this indeed belongs in another subforum.

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The point is that in classical GR, considered as an exact classical theory, without regard to whether it could be a low energy "effective theory" approximation to something else, gravity is simply not a force. Any view that considers gravity a force has to rely on some claim about GR being an "effective field theory" approximation to something else. Such claims are off topic in this particular forum because this particular forum is not about GR as a possible "effective field theory" approximation to something else; it's about classical GR, considered as an exact classical theory.
This seems to be a misunderstanding. Of course, I'd not use the word "force" at all in relativistic physics, because all the successful models of interactions are local field theories, i.e., there are no forces in the Newtonian sense, where they are described as actions at a distance. Nevertheless gravitation (which is synonymous to "gravitational interaction") is described as an interaction, by gauging Poincare symmetry, leading to the introduction of fields which can then be (re)interpreted as a connection and fundamental form of the spacetime manifold. See, e.g.,

T. W. B. Kibble, Lorentz Invariance and the Gravitational
Field, Jour. Math. Phys. 2, 212 (1960),
https://doi.org/10.1063/1.1703702

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GR is a relativistic classical field theory of the gravitational interaction.
Not if you allow spacetimes that don't have the same conformal structure at infinity as flat Minkowski spacetime, as I have already pointed out. For example, you can't explain FRW spacetime using this interpretation of GR.

It's ironic that you fight for a geometrical-interpretation-only point of view and at the same time forbid the discussion of a very strong argument for this point of view: Starting from the gauge-theoretical approach (making Poincare transformations local) you end up inevitably with the geometrical point of view.
Yes, this line of research was well explored in the 1960s and early 1970s, and they found, as you say, that when you construct the classical gauge theory of a massless spin-2 field correct to all orders, you end up with the Einstein-Hilbert Lagrangian and the Einstein Field Equation. But still with the limitation I have already described: you did all this on a flat background spacetime, and not all solutions of the EFE can be interpreted as fields on a flat background spacetime.

So this approach cannot be interpreted as a classical field theory that is fully equivalent to the standard geometric view of GR. Nor have any of its proponents claimed it was: as I have already noted, their rationale for taking this approach seriously has always been that it allows GR to be interpreted as a low energy "effective field theory" and motivates the search for a more fundamental theory of quantum gravity.

gravitation (which is synonymous to "gravitational interaction") is described as an interaction, by gauging Poincare symmetry, leading to the introduction of fields which can then be (re)interpreted as a connection and fundamental form of the spacetime manifold.

But still with the limitation I described above.

martinbn and Motore
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Not if you allow spacetimes that don't have the same conformal structure at infinity as flat Minkowski spacetime, as I have already pointed out. For example, you can't explain FRW spacetime using this interpretation of GR.
As I said, you end up with GR (or Einstein-Cartan theory), and it is a relativistic field theory. That there are solutions which are not asymptotically a Minkowski space is no contradiction but a solution of the field equations. Poincare invariance is only local (in the geometric approach this follows from the unique action assuming the lowest order of field derivatives in the action and the Einstein field equations; in the physical approach it's used as a gauge principle, and the geometric interpretation is derived). There is no contradiction between the geometric and the gauge-theoretical physical approach.

As I said, you end up with GR (or Einstein-Cartan theory), and it is a relativistic field theory. That there are solutions which are not asymptotically a Minkowski space is no contradiction but a solution of the field equations. Poincare invariance is only local (in the geometric approach this follows from the unique action assuming the lowest order of field derivatives in the action and the Einstein field equations; in the physical approach it's used as a gauge principle, and the geometric interpretation is derived). There is no contradiction between the geometric and the gauge-theoretical physical approach.
I think the point is that in the spin2 field approach you don't get all the solutions you have in GR, so in that sense they are not equvalent.

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Why don't you get the same solutions to the same equations, only because you derive them in a different way? The only thing that's different is that you get, if considering also matter fields with spin, an Einstein-Cartan rather than a pseudo-Riemannian (Lorentzian) spacetime manifold. For the usual "macroscopic" applications you get the latter and thus standard GR. There's nothing forbidding any of the solutions of GR only because you use the gauge principle and a field-theoretical approach as for all other interactions. The only thing that hasn't been achieved yet is a satisfactory quantization of the gravitational field.

Why don't you get the same solutions to the same equations, only because you derive them in a different way? The only thing that's different is that you get, if considering also matter fields with spin, an Einstein-Cartan rather than a pseudo-Riemannian (Lorentzian) spacetime manifold. For the usual "macroscopic" applications you get the latter and thus standard GR. There's nothing forbidding any of the solutions of GR only because you use the gauge principle and a field-theoretical approach as for all other interactions. The only thing that hasn't been achieved yet is a satisfactory quantization of the gravitational field.
Becasue the equations are local and they constrain only the metric, and not the global topology. If you consider equations of a field on ##\mathbb R^4##, how are you going to get a solution which has as an underlying manifold something else than ##\mathbb R^4##?

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Because I make the Poincare symmetry local. That's why you do not only get solutions, which are asymptotically Minkowski spacetime as solutions to GR.

Because I make the Poincare symmetry local. That's why you do not only get solutions, which are asymptotically Minkowski spacetime as solutions to GR.
Then I must have missunderstood which approach you meant. What is the gauge theoretical approach?

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You make the Poincare symmetry of special relativity/Minkowski space a local symmetry. A nice introduction to the formalism can be found in

P. Ramond, Quantum field theory - a modern primer, 2nd edition

where the corresponding section is about classical field theory only.

Two nicely readable original papers are

R. Utiyama, Invariant theoretical interpretation of
interaction, Phys. Rev. 101, 1597 (1956),
https://doi.org/10.1103/PhysRev.101.1597

T. W. B. Kibble, Lorentz Invariance and the Gravitational
Field, Jour. Math. Phys. 2, 212 (1960),
https://doi.org/10.1063/1.1703702

Dale
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That there are solutions which are not asymptotically a Minkowski space is no contradiction but a solution of the field equations.
You are not addressing my actual point. In order to do the construction, you start out with flat Minkowski spacetime. So you can't possibly end up with any spacetime that doesn't have the same conformal structure as flat Minkowski spacetime. It is no answer to that to point out that such spacetimes (e.g., FRW spacetime) are solutions of the field equations, because in the spin-2 field approach you don't just have the field equations, you got the field equations by a particular construction, so you can only justify it for solutions that are consistent with that construction.

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Two nicely readable original papers are
These are both paywalled so I can only read the abstracts, but it appears that at least in the second paper, the theory obtained is not the same as GR when matter is present: the Christoffel symbols are no longer symmetric in their lower indexes. Any such theory is speculative and off topic in this thread and this forum (we could discuss it in the Beyond the Standard Model forum).

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You are not addressing my actual point. In order to do the construction, you start out with flat Minkowski spacetime. So you can't possibly end up with any spacetime that doesn't have the same conformal structure as flat Minkowski spacetime. It is no answer to that to point out that such spacetimes (e.g., FRW spacetime) are solutions of the field equations, because in the spin-2 field approach you don't just have the field equations, you got the field equations by a particular construction, so you can only justify it for solutions that are consistent with that construction.
You start out with flat Minkowski space, or rather Poincare symmetry of this spacetime and make this symmetry local. This explicitly extends spacetime to something else, and this something else is Einstein-Cartan spacetime or, as a special case, the Lorentzian spacetime of GR. I don't see, why you think that in this approach you are restricted to solutions that are asymptotically Minkowski, ruling out solutions like FLRW, which are of course of high physical significance (cosmology).

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These are both paywalled so I can only read the abstracts, but it appears that at least in the second paper, the theory obtained is not the same as GR when matter is present: the Christoffel symbols are no longer symmetric in their lower indexes. Any such theory is speculative and off topic in this thread and this forum (we could discuss it in the Beyond the Standard Model forum).
Yes, that's why I say in the most general case of matter fields with arbitrary spin you get a Einstein Cartan manifold rather than a Lorentzian (GR) one. That's not a contradiction to the success of GR in describing all observations, which deal with spin-saturated macroscopic matter and the em. field only. In this case you deal with scalar fields only (to describe matter) and the em. field, which is a gauge field (gauging the U(1) symmetry of electrodynamics), which due to this special nature as a gauge field, doesn't cause torsion.

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You start out with flat Minkowski space, or rather Poincare symmetry of this spacetime and make this symmetry local. This explicitly extends spacetime to something else
If you have to start out with a global flat spacetime, I'm not sure how you can "extend" it to something else; Minkowski spacetime is already its own maximal extension. Putting a field on it can't change that.

If, OTOH, you mean you don't start with any global spacetime, but just with the concept of "local Poincare symmetry", that would be different; the potential issue I see there is how you even define "local Poincare symmetry" without an underlying manifold. Note that in other cases where a gauge symmetry is made local, such as electromagnetism, you already have an underlying manifold, namely Minkowski spacetime; the gauge symmetry process doesn't affect the manifold or its metric. In the case you're describing, though, it does.

Is there a non-paywalled version of either of the papers you referenced?

I looked at Kibble and Ramond, and as far as I can tell they are not addressing @PeterDonis 's question. They work locally and do not say anything about the global properties of the space-time.

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If you have to start out with a global flat spacetime, I'm not sure how you can "extend" it to something else; Minkowski spacetime is already its own maximal extension. Putting a field on it can't change that.

If, OTOH, you mean you don't start with any global spacetime, but just with the concept of "local Poincare symmetry", that would be different; the potential issue I see there is how you even define "local Poincare symmetry" without an underlying manifold. Note that in other cases where a gauge symmetry is made local, such as electromagnetism, you already have an underlying manifold, namely Minkowski spacetime; the gauge symmetry process doesn't affect the manifold or its metric. In the case you're describing, though, it does.

Is there a non-paywalled version of either of the papers you referenced?
The idea is to start with a symmetry group and construct the "geometry" following from this group. You start with some action whose 1st variation has a given global Lie symmetry and make this symmetry local.

That's amazingly successful for all the interactions, and gravity is no exception. For me it was the eye opener, why gravity has to do with spacetime symmetry: It's because the unique feature for this interaction it's the symmetry group of spacetime which is "gauged". This leads to the introduction of tetrades and a connection with in general torsion. The torsion is related to spin. An exception are gauge-vector fields (like the em. field), which provide no torsion. Since also the classical description of matter (continuum mechanics) is described by scalar fields, for the usual "macroscopic applications" you thus end up with GR.

The idea is to start with a symmetry group and construct the "geometry" following from this group. You start with some action whose 1st variation has a given global Lie symmetry and make this symmetry local.

That's amazingly successful for all the interactions, and gravity is no exception. For me it was the eye opener, why gravity has to do with spacetime symmetry: It's because the unique feature for this interaction it's the symmetry group of spacetime which is "gauged". This leads to the introduction of tetrades and a connection with in general torsion. The torsion is related to spin. An exception are gauge-vector fields (like the em. field), which provide no torsion. Since also the classical description of matter (continuum mechanics) is described by scalar fields, for the usual "macroscopic applications" you thus end up with GR.
How do you get the global structure of the space-time? I suppose I need to look at your references in detail, but I didn't see anything in that respect. Is it possible to give an example? Say Minkowski spacetime, which is ##\mathbb R^4## with the usual metric, and a flat torus, which is ##\mathbb R^4 / \mathbb Z^4## with the same metric. Locally they are the same, but they are different space-times in GR.

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You start with some action whose 1st variation has a given global Lie symmetry and make this symmetry local.
I don't see how it's possible to do this without already having an underlying manifold defined, since without one the terms "global" and "local" have no meaning.

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How do you get the global structure of the space-time? I suppose I need to look at your references in detail, but I didn't see anything in that respect. Is it possible to give an example? Say Minkowski spacetime, which is ##\mathbb R^4## with the usual metric, and a flat torus, which is ##\mathbb R^4 / \mathbb Z^4## with the same metric. Locally they are the same, but they are different space-times in GR.
I don't know, whether there is any way to determine the global structure of spacetime since we can observe only a pretty local part of the universe. It's maybe a philosophical question, what the global structure of spacetime might be.

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I don't see how it's possible to do this without already having an underlying manifold defined, since without one the terms "global" and "local" have no meaning.
For this application you start with a special relativistic Lagrangian. The most simple case is that for a scalar field ("Klein-Gordon field"),
$$\mathcal{L}=\frac{1}{2} (\partial_{\mu} \phi) (\partial^{\mu} \phi)-\frac{m^2}{2} -V(\phi).$$
Under Poincare transformations the field transforms as
$$\phi'(x')=\phi(x)=\phi[\Lambda^{-1} (x'-a)]$$
with ##\Lambda## a constant ##\text{SO}(3,1)^{\uparrow}## matrix and ##a## a constant four-vector.

To "make that symmetry local" means to make ##\Lambda## and ##a## dependent on ##x## and introduce appropriate "connections" for "covariant derivatives" such that the Lagrangian (or rather the first variation of the action) stays invariant.

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I don't know, whether there is any way to determine the global structure of spacetime since we can observe only a pretty local part of the universe.
We're talking about a particular model. If that model requires assuming a particular underlying manifold, it can't possibly have valid solutions that have a different underlying manifold. That is true regardless of our ability to test the model against observations. However, if our best current model of the universe as a whole has a different underlying manifold than the one your model requires, that at least is a strong indicator that your model's limitations might be important.

Which means you're assuming flat Minkowski spacetime as your underlying manifold, so, again, your model can't possibly have valid solutions, such as FRW spacetime, that aren't compatible with that underlying manifold. So, again, if our best current model of the universe as a whole uses FRW spacetime, that at least is a strong suggestion that your model is not a valid model for the universe as a whole. And that, to me, is a serious limitation.

To "make that symmetry local" means to make ##\Lambda## and ##a## dependent on ##x##
Yes, and what is ##x##? It's a label for points in the underlying manifold. So you already have to have an underlying manifold for this to have any meaning. And, as above, the underlying manifold in this model is Minkowski spacetime.

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If you change the spacetime symmetry from "global" to "local" you introduce a more general spacetime manifold. It's really too lengthy to post the details here. See the paper's I've sent you via PM.

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If you change the spacetime symmetry from "global" to "local" you introduce a more general spacetime manifold. It's really too lengthy to post the details here. See the paper's I've sent you via PM.
I have looked through the papers you sent, and I do not see where "introduce a more general spacetime manifold" is being done in either one. I see that the spin-2 field is being interpreted as a correction to the metric, but that doesn't change the underlying manifold, it just changes the metric on it.

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Making the spacetime symmetry local changes of course the spacetime structure itself. It's no longer an affine Lorentzian manifold anymore but a more general one. It turns out that the newly introduced fields can be reinterpreted as connections (most generally with torsion) and a Lorentzian metric (via the tetrades, ##g_{\mu \nu}=\eta_{ab} e_{\mu}^a e_{\nu}^b##).

In the geometric approach you start with the spacetime as a differential manifold with a connection and a metric (in standard GR a pseudo-Riemannian manifold, with the usual unique metric compatible torsion-free connection). Then the matter fields are introduced via the corresponding action coupled to gravity via the metric.

In the gauge-theoretical approach you start with the global Poincare symmetry on Minkowski space and a model for the matter (e.g., for a relativistic fluid, which can be represented with three scalar fields and the em. field, which is a gauge field) and introduce the gravitational interaction by making the Poincare symmetry local (which implies general coordinates under diffeomorphisms). It turns then out that you can reinterpret the introduced "gauge fields" (tetrades and connections) to make this symmetry local as a metric and connections. The "matter content" of the model determines whether you get a Einstein-Cartan manifold with torsion (coupling to spin) or a pseudo-Riemannian (Lorentzian) manifold.

Dale
Making the spacetime symmetry local changes of course the spacetime structure itself. It's no longer an affine Lorentzian manifold anymore but a more general one. It turns out that the newly introduced fields can be reinterpreted as connections (most generally with torsion) and a Lorentzian metric (via the tetrades, ##g_{\mu \nu}=\eta_{ab} e_{\mu}^a e_{\nu}^b##).

In the geometric approach you start with the spacetime as a differential manifold with a connection and a metric (in standard GR a pseudo-Riemannian manifold, with the usual unique metric compatible torsion-free connection). Then the matter fields are introduced via the corresponding action coupled to gravity via the metric.

In the gauge-theoretical approach you start with the global Poincare symmetry on Minkowski space and a model for the matter (e.g., for a relativistic fluid, which can be represented with three scalar fields and the em. field, which is a gauge field) and introduce the gravitational interaction by making the Poincare symmetry local (which implies general coordinates under diffeomorphisms). It turns then out that you can reinterpret the introduced "gauge fields" (tetrades and connections) to make this symmetry local as a metric and connections. The "matter content" of the model determines whether you get a Einstein-Cartan manifold with torsion (coupling to spin) or a pseudo-Riemannian (Lorentzian) manifold.
But you say it yourself. The underlying manifold is Minkowski space. No matter what additional structure you have the manifold is still Minkowski space. In GR the underlying manifold can be something else, for example you can have a compact manifold (may not be physically relevant, but it is allowed). This means that the two theories cannot be equivalent.

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But you end up with the same equations in both approaches. The spacetime manifold with gravity is not an affine space, i.e., there are no global inertial frames but only local ones.

But you end up with the same equations in both approaches. The spacetime manifold with gravity is not an affine space, i.e., there are no global inertial frames but only local ones.
The equations do not determine the manifold. Frames have no relevanve here.

General relativity says that spacetime is a four dimensional manifold ##M## with a metric ##g## which satisfies Einstein's equation. What you discribe above says that spacetime is Minkowski space plus the addisional fields and symetries, and you can interpret something as a metric sarisfying the same equations. The manifolds ##M## and Minkowski need not be the same.

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The usual way you derive gauge field theories gauging the Poincare symmetry (i.e., making it a local symmetry) leads to precisely what you define as General relativity.

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Making the spacetime symmetry local changes of course the spacetime structure itself. It's no longer an affine Lorentzian manifold anymore but a more general one.
You gave references to two papers. I don't see this claim either made or justified in either one of them. All I see the papers doing is showing that the spin-2 gauge field can be interpreted as a correction to the metric. That doesn't change the underlying manifold, which remains ##R^4##. It only changes the metric on the manifold. But many solutions to the Einstein Field Equation do not have ##R^4## as their underlying manifold.

If you think those papers do make and justify the stronger claim that making the spacetime symmetry local can change the underlying manifold, please give me some specific pointers to where in the papers this is done.

weirdoguy
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This means that the two theories cannot be equivalent.
Well, there is a large class of physical solutions in common, but there are some solutions in one that are not in the other. The class of common solutions includes all local solutions and some global ones.

dextercioby
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I don't understand, why you think the resulting theory of gravity from the gauge principle is different from GR (it's even more general, because it includes also torsion in the case of matter with spin). You get the very same equations as in GR, only the derivation is different, and the spacetime cannot be Minkowski space, because the curvature is not 0. For me it's obvious that this is the same physical theory as (extended) GR.

It's also explicitly mentioned in Sect. 6 of Kibbles paper:

T. W. B. Kibble, Lorentz Invariance and the Gravitational
Field, Jour. Math. Phys. 2, 212 (1960),
https://doi.org/10.1063/1.1703702

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I don't understand, why you think the resulting theory of gravity from the gauge principle is different from GR
A very specific concern has been raised in this thread, which you have not addressed. We all agree that the field equation of the theory of gravity derived from the gauge principle is the Einstein Field Equation. That's not an issue. The issue is that the underlying manifold (not the metric but the underlying manifold) that is permitted by the gauge theory construction, as far as we can tell, must be ##R^4##, whereas GR considered as a geometric theory of gravity without any gauge theory justification admits other underlying manifolds.

So far your only response to this concern has been "the gauge theory construction allows other underlying manifolds, but it's too complicated to explain how here, look at these papers". As I have already posted, I don't see anything in the papers you referenced that justifies the claim that other underlying manifolds than ##R^4## are permitted.

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I don't understand, why you think the resulting theory of gravity from the gauge principle is different from GR (it's even more general, because it includes also torsion in the case of matter with spin).
In GR you start with a pseudo-Riemannian manifold which is a topological manifold equipped with a metric (with a -+++ signature). That topological manifold is arbitrary and may have highly nontrivial global topology. However, locally, in a neighborhood around any event in the manifold you have that simple topology.

If you start with an underlying Lorentzian manifold and construct gauge fields on top of that Lorentzian manifold then your global topology is inherited from the underlying manifold. You can only have the trivial global topology by construction.

You get the very same equations as in GR, only the derivation is different
You get the same equations as in GR (neglecting matter with spin), but those equations are local equations. They don’t inform the global topology. That global topology must be constructed in some other manner.

In GR you can use any topological manifold. But in the gauge approach the topology is necessarily trivial by construction.

Frankly, I don’t think this is a particular weakness. I don’t think that we have any evidence that the physical spacetime of the universe has non-trivial topology.

vanhees71 and weirdoguy
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